1# Light and matter 2 3## Photoelectric effect 4 5### Planck's equation 6 7$$E=hf,\quad f={c \over \lambda}$$ 8$$\therefore E={hc \over \lambda}$$ 9 10where 11$E$ is energy of a quantum of light (J) 12$f$ is frequency of EM radiation 13$h$ is Planck's constant ($6.63\times 10^{-34}\operatorname{J s}=4.12\times 10^{-15} \operatorname{eV s}$) 14 15### Electron-volts 16 17$$ 1 \operatorname{eV} = 1.6\times 10^{-19} \operatorname{J}$$ 18 19*Amount of energy an electron gains when it moves through a potential difference of 1V* 20 21- equivalent unit is Joule seconds (e.g. $h$) 22 23### Photoelectric effect 24 25- some metals becomes positively charged when hit with EM radiation 26- this is due to e- being ejected from surface of metal 27- *photocurrent* - flow of e- due to photoelectric effect 28- causes increase in current in a circuit 29- $V_{\operatorname{supply}}$ does not affect photocurrent 30- if $V_{\operatorname{supply}} \gt 0$, e- are attracted to collector anode. 31- if $V_{\operatorname{supply}} \lt 0$, e- are attracted to illuminated cathode, and $I\rightarrow 0$ 32- not all electrons have the same velocity - depends on ionisation energy (shell) 33 34#### Wave / particle (quantum) models 35wave model: 36 37- cannot explain photoelectric effect 38- $f$ is irrelevant to photocurrent 39- predicts that there should be a delay between incidence of radiation and ejection of e- 40 41particle model: 42 43- explains photoelectric effect 44- rate of photoelectron release is proportional to intensity of incident light 45- shining light on a metal "bombards" it with photons 46- no time delay 47 48#### Work function and threshold frequency 49 50- *threshold frequency* $f_0$ - minimum frequency for photoelectrons to be ejected 51- if $f \lt f_0$, no photoelectrons are detected 52 53- Einstein: energy required to eject photoelectron is constant for each metal 54- *work function* $\phi$ - minimum energy required to release photoelectrons 55- $\phi$ is determined by strength of bonding 56 57$$\phi=hf_0$$ 58 59#### $E_K$ of photoelectrons (stopping energy) 60 61$$E_{\operatorname{k-max}}=hf - \phi$$ 62 63where 64$E_k$ is max energy of an emmitted photoelectron 65$f$ is frequency of incident photon (**not** emitted electron) 66$\phi$ is work function ("latent" energy) 67 68Gradient of a frequency-energy graph is equal to $h$ 69y-intercept is equal to $\phi$ 70 71#### Stopping potential $V_0$ 72$$V_0 = {E_{K \operatorname{max}} \over q_e} = {{hf - \phi} \over q_e}$$ 73 74## Wave-particle duality 75 76### Double slit experiment 77Particle model allows potential for photons to interact as they pass through slits. However, an interference pattern still appears when a dim light source is used so that only one photon can pass at a time. 78 79## De Broglie's theory 80 81$$\lambda = {h \over \rho} = {h \over mv}$$ 82 83- theorised that matter may display both wave- and particle-like properties like light 84- predict wavelength of a particle with $\lambda = {h \over \rho}$ where $\rho = mv$ 85- impossible to confirm de Broglie's theory of matter with double-slit experiment, since wavelengths are much smaller than for light, requiring an equally small slit ($< r_{\operatorname{proton}}$) 86- confirmed by Davisson and Germer's apparatus (diffraction pattern like double-slit) 87- also confirmed by Thomson - e- diffraction pattern resembles x-ray (wave) pattern 88- electron is only stable in orbit if $mvr = n{h \over 2\pi}$ where $n \in \mathbb{Z}$ 89- rearranging this, $2\pi r = n{h \over mv}$ (circumference) 90- therefore, stable orbits are those where circumference = whole number of e- wavelengths 91- if $2\pi r \ne n{h \over mv}$, interference occurs when pattern is looped and standing wave cannot be established 92 93### Photon momentum 94 95$$\rho = {hf \over c} = {h \over \lambda}$$ 96- if a massy particle (e.g. electron) has a wavelength, then anything with a wavelength must have momentum 97- therefore photons have (theoretical) momentum 98- to solve photon momentum, rearrange $\lambda = {h \over mv}$ 99 100## Spectral analysis 101 102 103### Absorption 104- Black lines in spectrum show light not reflected 105 106### Emission 107- Coloured lines show light being ejected from e- shells 108- Energy change between ground / excited state: $\Delta E = hf = {hc \over \lambda}$ 109- Bohr's model describes discrete energy levels 110- Energy is conserved (out = in) 111- Ionisation energy - minimum energy required to remove an electron 112- EMR is absorbed/emitted when $E_{\operatorname{K-in}}=\Delta E_{\operatorname{shells}}$ (i.e. $\lambda = {hc \over \Delta E_{\operatorname{shells}}}$) 113 114## Light sources 115- **incandescent:** <10% efficient, broad spectrum 116- **LED:** semiconducting doped-Si diodes 117- - most electrons in *valence band* (energy level) 118- - provided energy, electrons can jump to *conduction band* and move through Si as current 119- - colour determined by $\Delta E$ between bands (shells), and type of doping 120- **laser:** gas atoms are excited 121- - *popular inversion* - most gas atoms are excited 122- - photons are released if stimulated by another photon of the right wavelength 123- **synchrotron:** - magnetically accelerates electrons 124- - extremely bright 125- - highly polarised 126- - emitted in short pulses 127- - broad spectrum 128 129## Quantum mechanics 130 131- uncertainty occurs in any measurement 132- inherent physical limit to absolute accuracy of measurements (result of wave-particle duality) 133- interaction between observer and object 134- measuring location of an e- requires hitting it with a photon, but this causes $\rho$ to be transferred to electron, moving it 135 136### Indeterminancy principle 137 138$$\sigma E \sigma t \ge {h \over 4 \pi}$$ 139 140where $\sigma n$ is the uncertainty of $n$ 141 142**$\sigma E$ and $\sigma t$ are inversely proportional$** 143 144Therefore, position and velocity cannot simultaneously be known with 100% certainty. 145 146### Single-slit diffraction 147 148- one photon passes through slit at any time (controlled by intensity) 149- diffraction pattern can be explained by wave front split into wavelets 150- diffraction can be represented as uncertainty of photonic momentum 151 152 153### Comparison with Bohr's model 154 155**Newtonian (deterministic) model** - current $x$ and $v$ are known, so future $x$ can be calculated 156 157**Quantum mechanical model** - electron clouds rather than discrete shells (electrons are not particlces). We can only calculate probability of an electron being observed at a particular position 158 159 160 161774 abc melbourne